Graph Connections Relationships between Graph Theory and Other Areas of Mathematics

, by Beineke, Lowell Wayne; Wilson, Robin James

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ISBN: 9780198514978 | 0198514972
Cover: Hardcover
Copyright: 4/10/1997

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This new book introduces mathematicians to the applicability of graph theory to other areas, from number theory to linear algebra, neural networks, and finance. This is achieved through a series of expository chapters, each written by an expert in a different field. Each chapter has been carefully edited so that the terminology and notation are as standardized as possible. The book will be useful to both graph theorists and practitioners in other areas.

1 Introduction

(12)

Robin J. Wilson

1.1 Graphs

(2)

1.2 Adjacency and incidence

(1)

1.3 Paths and cycles

(1)

1.4 New graphs from old

(2)

1.5 Examples of graphs

(2)

1.6 Planar graphs

(2)

1.7 Colouring graphs

(1)

1.8 The efficiency of algorithms

(1)

1.9 And finally ...

(1)

References

(1)

2 Enumeration

(21)

Ronald C. Read

2.1 Introduction

(1)

2.2 Labelled graphs and generating functions

(2)

2.3 Necklaces

(1)

2.4 Polya's Enumeration Theorem

(3)

2.5 Chemical enumeration

(4)

2.6 The enumeration of graphs

(3)

2.7 Connected graphs

(2)

2.8 Trees and rooted trees

(2)

2.9 Other kinds of graphs

(1)

2.10 Unsolved problems

(1)

References

(2)

3 Number Theory

(18)

Roger Cook

3.1 Introduction

(1)

3.2 Multiplicative functions

(1)

3.3 The Mobius function

(2)

3.4 Euler's function

(1)

3.5 Polya's Enumeration Theorem

(1)

3.6 Eulerian graphs and tournaments

(2)

3.7 Finite fields and Paley graphs

(2)

3.8 Quadratic residue tournaments

(1)

3.9 Hadamard matrices and designs

(1)

3.10 Ramanujan graphs

(2)

3.11 Negative Pell equations

(1)

References

(3)

4 Partial Orders

(18)

Graham Brightwell

4.1 Introduction

(1)

4.2 Preliminaries

(3)

4.3 Dilworth's Theorem

(3)

4.4 Comparability graphs

(2)

4.5 Covering graphs and diagrams

(1)

4.6 Schnyder's Theorem

(5)

4.7 The incidence order

(1)

References

(3)

5 First-order Logic

(16)

Peter J. Cameron

5.1 Introduction

(1)

5.2 First-order logic

(3)

5.3 First-order properties of graphs

(2)

5.4 Applications of compactness

(2)

5.5 The random graph

(1)

5.6 Homogeneous graphs

(2)

5.7 No-categorical graphs

(1)

5.8 Sparse graphs

(3)

References

(2)

6 Linear Algebra

(14)

Peter Rowlinson

6.1 Introduction

(1)

6.2 Graph spectra

(2)

6.3 Distance-regular graphs

(3)

6.4 Other algebraic invariants

(3)

6.5 Eigenvalues and star partitions

(3)

References

(2)

7 Matroids

100

(16)

James Oxley

7.1 Introduction

100

(1)

7.2 Definitions and examples

101

(2)

7.3 Basic matroid operations

103

(3)

7.4 Connectivity

106

(2)

7.5 Wheels and whirls

108

(1)

7.6 Minimally 3-connected graphs and matroids

109

(2)

7.7 Excluded minors

111

(2)

7.8 Infinite antichains

113

(1)

7.9 Conclusion

114

(1)

References

114

(2)

8 Codes

116

(12)

Robert T. Curtis

Tony R. Morris

8.1 Introduction

116

(1)

8.2 Self-dual doubly-even codes

116

(1)

8.3 Invariant theory

117

(2)

8.4 Constructing binary codes from graphs

119

(2)

8.5 How many codes arise like this?

121

(1)

8.6 Graphs with the required properties

122

(5)

8.7 Counting tetrads

127

(1)

References

127

(1)

9 Groups

128

(13)

Peter J. Cameron

9.1 Introduction

128

(1)

9.2 An example: the Petersen graph

128

(1)

9.3 Three kinds of groups

129

(1)

9.4 Universal classes

130

(2)

9.5 Bounds

132

(1)

9.6 Random graphs

133

(1)

9.7 Vertex-transitive graphs

134

(1)

9.8 Distance-transitive graphs

135

(2)

9.9 Local structure

137

(1)

9.10 Computational aspects

138

(1)

References

139

(2)

10 Geometry

141

(14)

Edward R. Scheinerman

10.1 Introduction

141

(1)

10.2 Dimension 0: discrete sets

142

(2)

10.3 Dimension 1: interval graphs

144

(3)

10.4 Dimension 1 and a little: trees and circles

147

(1)

10.5 Dimensions 2 and higher

148

(2)

10.6 Counting methods via real algebraic geometry

150

(3)

References

153

(2)

11 Topology

155

(21)

Lowell W. Beineke

11.1 Introduction

155

(1)

11.2 Planar graphs

156

(2)

11.3 Thickness

158

(2)

11.4 Crossing numbers

160

(4)

11.5 Orientable surfaces and rotation systems

164

(2)

11.6 Genus and chromatic numbers

166

(2)

11.7 Non-orientable surfaces

168

(3)

11.8 Kuratowski-type theorems

171

(2)

References

173

(3)

12 Knots

176

(18)

Dominic Welsh

12.1 Introduction

176

(1)

12.2 Basic concepts

177

(2)

12.3 Tait colourings

179

(2)

12.4 Classifying knots

181

(1)

12.5 Braids and the Seifert graph

182

(2)

12.6 The Jones and Kauffman bracket polynomials

184

(3)

12.7 Bivariate polynomials

187

(1)

12.8 The Tait conjectures

188

(2)

12.9 Two applications

190

(2)

References

192

(2)

13 Probability

194

(14)

Colin McDiarmid

13.1 Introduction

194

(1)

13.2 Random graphs: usual behaviour

194

(4)

13.3 Random graphs: deterministic results

198

(1)

13.4 The Lovasz Local Lemma

199

(2)

13.5 Deterministic graphs: random methods

201

(3)

13.6 Concentration of measure and isoperimetric inequalities

204

(2)

References

206

(2)

14 Statistics

208

(19)

Peter Wild

14.1 Introduction

208

(1)

14.2 Experimental design

209

(6)

14.3 A-optimality and closed walks

215

(3)

14.4 Bounds

218

(3)

14.5 Spanning trees and D-optimality

221

(1)

14.6 Line graphs and E-optimality

222

(2)

14.7 Row-column designs

224

(1)

14.8 Conclusion

225

(1)

References

225

(2)

15 Computing

227

(20)

Robin Whitty

15.1 Introduction

227

(1)

15.2 Languages

228

(1)

15.3 Grammars

228

(3)

15.4 Finite automata

231

(4)

15.5 Flowgraphs

235

(4)

15.6 Program analysis

239

(5)

References

244

(3)

16 Artificial Neural Networks

247

(14)

Martin Anthony

16.1 Introduction

247

(1)

16.2 Artificial neural networks

248

(1)

16.3 Boltzmann machines

248

(2)

16.4 Optimization with Boltzmann machines

250

(2)

16.5 Feedforward networks

252

(1)

16.6 Supervised learning in feedforward networks

253

(6)

References

259

(2)

17 International Finance

261

(19)

Norman Biggs

17.1 Introduction

261

(1)

17.2 Exchange dealing then and now

262

(2)

17.3 Exchange rate networks

264

(2)

17.4 Some classical theory

266

(2)

17.5 The export and import points

268

(3)

17.6 Potential theory

271

(2)

17.7 Determination of exchange rates by cash flows

273

(4)

17.8 Application to a tournament ranking problem

277

(1)

17.9 Mechanisms linking exchange rates and trade

277

(1)

References

278

(2)

Notes on Contributors

280

(5)

Index

285

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Graph Connections Relationships between Graph Theory and Other Areas of Mathematics

Summary

Table of Contents

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Graph Connections Relationships between Graph Theory and Other Areas of Mathematics

Summary

Table of Contents

Supplemental Materials

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